Partially cable-stayed bridges have the characteristics of continuous rigid-frame bridges and cable-stayed bridges, making them a novel composite bridge system. This study focuses on the construction project of a multi-tower high-pier curved partially cable-stayed bridge to investigate the bridge’s stability during construction. The Midas/Civil software was used to establish a model for key construction stages of the bridge, considering structural linear elasticity and geometric nonlinearity. The study examines the impact of static wind loads, asymmetric construction of the main girder, closure sequence, and the load and detachment of the hanging basket on the bridge’s stability during construction. The results indicate that static wind loads have a significant impact on structural geometric nonlinearity, with a maximum reduction of 4.99%. Asymmetric construction at both ends of the main girder can cause structural instability and should be avoided. The geometric nonlinearity stability coefficient for the hanging basket load decreased by 10.83% during the maximum no-cable stage and by 7.84% during the cable stage, significantly affecting the stability during construction. A bridge closure sequence of side-span, secondary midspan, and midspan provides the most stable condition during the construction phase. The results of this study can inform the construction of similar partially cable-stayed bridges.